Introduction to

EngineeringMechanics

A Continuum Approach

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CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

Jenn Stroud RossmannLafayette College

Easton, Pennsylvania, USA

Clive L. DymHarvey Mudd College

Claremont, California, USA

Introduction to

EngineeringMechanics

A Continuum Approach

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

© 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1

International Standard Book Number-13: 978-1-4200-6271-7 (Hardcover)

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Library of Congress Cataloging-in-Publication Data

Rossman, Jenn Stroud.Introduction to engineering mechanics: A continuum approach / Jenn Stroud

Rossman, Clive L. Dym.p. cm.

Includes bibliographical references and index.ISBN 978-1-4200-6271-7 (alk. paper)1. Mechanics, Applied. I. Dym, Clive L. II. Title.

TA350.B348 1986620.1--dc22 2008033432

Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.com

and the CRC Press Web site athttp://www.crcpress.com

v

Contents

Preface .........................................................................................................xvAbout the Authors ................................................................................... xvii

1 Introduction......................................................................................... 11.1 A Motivating Example: Remodeling an Underwater Structure ....21.2 Newton’s Laws: The First Principles of Mechanics .........................41.3 Equilibrium ...........................................................................................51.4 Definition of a Continuum ..................................................................61.5 Mathematical Basics: Scalars and Vectors .........................................91.6 Problem Solving .................................................................................. 121.7 Examples .............................................................................................. 13

Example 1.1 .......................................................................................... 13Solution ..................................................................................... 13

Example 1.2 .......................................................................................... 15Solution ..................................................................................... 16

1.8 Problems .............................................................................................. 17Notes ............................................................................................................. 18

2 Strain and Stress in One Dimension ............................................. 192.1 Kinematics: Strain ............................................................................... 20

2.1.1 Normal Strain .......................................................................... 202.1.2 Shear Strain ..............................................................................232.1.3 Measurement of Strain ........................................................... 24

2.2 The Method of Sections and Stress ..................................................252.2.1 Normal Stresses ...................................................................... 272.2.2 Shear Stresses ..........................................................................28

2.3 Stress–Strain Relationships ............................................................... 322.4 Equilibrium .........................................................................................362.5 Stress in Axially Loaded Bars ........................................................... 372.6 Deformation of Axially Loaded Bars ...............................................402.7 Equilibrium of an Axially Loaded Bar ............................................422.8 Indeterminate Bars .............................................................................43

2.8.1 Force (Flexibility) Method .....................................................442.8.2 Displacement (Stiffness) Method..........................................46

2.9 Thermal Effects ...................................................................................482.10 Saint-Venant’s Principle and Stress Concentrations ...................... 492.11 Strain Energy in One Dimension ..................................................... 512.12 A Road Map for Strength of Materials ............................................532.13 Examples ..............................................................................................55

Example 2.1 ..........................................................................................55Solution .....................................................................................55

vi Introduction to Engineering Mechanics: A Continuum Approach

Example 2.2 ..........................................................................................56Solution ..................................................................................... 57

Example 2.3 .......................................................................................... 57Solution .....................................................................................58

Example 2.4 .......................................................................................... 59Solution ..................................................................................... 59

Example 2.5 ..........................................................................................60Solution ..................................................................................... 61

Example 2.6 .......................................................................................... 62Solution ..................................................................................... 62

Example 2.7 ..........................................................................................64Solution .....................................................................................65

Example 2.8 ..........................................................................................66Solution .....................................................................................66

Example 2.9 .......................................................................................... 67Solution .....................................................................................68

2.14 Problems .............................................................................................. 69Case Study 1: Collapse of the Kansas City Hyatt Regency Walkways ............................................................................................. 76

Problems .............................................................................................. 82Notes ............................................................................................................. 82

3 Strain and Stress in Higher Dimensions ...................................... 853.1 Poisson’s Ratio .....................................................................................853.2 The Strain Tensor ................................................................................ 873.3 Strain as Relative Displacement .......................................................903.4 The Stress Tensor ................................................................................ 923.5 Generalized Hooke’s Law .................................................................. 963.6 Limiting Behavior ............................................................................... 973.7 Properties of Engineering Materials .............................................. 101

Ferrous Metals ................................................................................... 103Nonferrous Metals............................................................................ 103Nonmetals.......................................................................................... 104

3.8 Equilibrium ....................................................................................... 1043.8.1 Equilibrium Equations ......................................................... 1053.8.2 The Two-Dimensional State of Plane Stress ...................... 1073.8.3 The Two-Dimensional State of Plane Strain ..................... 108

3.9 Formulating Two-Dimensional Elasticity Problems ................... 1093.9.1 Equilibrium Expressed in Terms of Displacements ......... 1103.9.2 Compatibility Expressed in Terms of Stress Functions ... 1113.9.3 Some Remaining Pieces of the Puzzle of General Formulations ..................................................................................... 112

3.10 Examples ............................................................................................ 114Example 3.1 ........................................................................................ 114

Solution ................................................................................... 115Example 3.2 ........................................................................................ 116

Contents vii

Solution ................................................................................... 1163.11 Problems ............................................................................................. 116Notes ........................................................................................................... 121

4 Applying Strain and Stress in Multiple Dimensions ................ 1234.1 Torsion ................................................................................................ 123

4.1.1 Method of Sections ................................................................ 1234.1.2 Torsional Shear Stress: Angle of Twist and the Torsion Formula ................................................................................ 1254.1.3 Stress Concentrations ........................................................... 1304.1.4 Transmission of Power by a Shaft ....................................... 1314.1.5 Statically Indeterminate Problems ..................................... 1324.1.6 Torsion of Inelastic Circular Members ............................... 1334.1.7 Torsion of Solid Noncircular Members .............................. 1354.1.8 Torsion of Thin-Walled Tubes ............................................. 138

4.2 Pressure Vessels ................................................................................ 1414.3 Transformation of Stress and Strain .............................................. 145

4.3.1 Transformation of Plane Stress ........................................... 1464.3.2 Principal and Maximum Stresses....................................... 1494.3.3 Mohr’s Circle for Plane Stress ............................................. 1514.3.4 Transformation of Plane Strain ........................................... 1544.3.5 Three-Dimensional State of Stress ..................................... 156

4.4 Failure Prediction Criteria ............................................................... 1574.4.1 Failure Criteria for Brittle Materials ................................... 158

4.4.1.1 Maximum Normal Stress Criterion ..................... 1584.4.1.2 Mohr’s Criterion ...................................................... 159

4.4.2 Yield Criteria for Ductile Materials .................................... 1614.4.2.1 Maximum Shearing Stress (Tresca) Criterion .... 1614.4.2.2 Von Mises Criterion ............................................... 162

4.5 Examples ............................................................................................ 162Example 4.1 ........................................................................................ 162

Solution ................................................................................... 163Example 4.2 ........................................................................................ 163

Solution ................................................................................... 163Example 4.3 ........................................................................................ 165

Solution ................................................................................... 165Example 4.4 ........................................................................................ 165

Solution ................................................................................... 165Example 4.5 ........................................................................................ 166

Solution ................................................................................... 166Example 4.6 ........................................................................................ 168

Solution ................................................................................... 168Example 4.7 ........................................................................................ 170

Solution ................................................................................... 170Example 4.8 ........................................................................................ 171

Solution ................................................................................... 171

viii Introduction to Engineering Mechanics: A Continuum Approach

Example 4.9 ........................................................................................ 172Solution ................................................................................... 172

Example 4.10 ...................................................................................... 177Solution ................................................................................... 177

Example 4.11 ...................................................................................... 180Solution ................................................................................... 180

4.6 Problems ............................................................................................ 183Case Study 2: Pressure Vessel Safety ..................................................... 188

Why Are Pressure Vessels Spheres and Cylinders? .................... 189Why Do Pressure Vessels Fail? ....................................................... 194Problems ............................................................................................ 197

Notes ...........................................................................................................200

5 Beams ............................................................................................... 2015.1 Calculation of Reactions .................................................................. 2015.2 Method of Sections: Axial Force, Shear, Bending Moment ........ 202

Axial Force in Beams........................................................................ 203Shear in Beams .................................................................................. 203Bending Moment in Beams ............................................................. 205

5.3 Shear and Bending Moment Diagrams ......................................... 206Rules and Regulations for Shear and Bending Moment Diagrams ............................................................................................ 206

Shear Diagrams ..................................................................... 206Moment Diagrams ................................................................ 207

5.4 Integration Methods for Shear and Bending Moment ................ 2075.5 Normal Stresses in Beams ............................................................... 2105.6 Shear Stresses in Beams ................................................................... 2145.7 Examples ............................................................................................ 221

Example 5.1 ........................................................................................ 221Solution ................................................................................... 221

Example 5.2 ........................................................................................223Solution ................................................................................... 224

Example 5.3 ........................................................................................229Solution ...................................................................................230

Example 5.4 ........................................................................................ 231Solution ................................................................................... 232

Example 5.5 ........................................................................................234Solution ...................................................................................235

Example 5.6 ........................................................................................236Solution ................................................................................... 237

5.8 Problems ............................................................................................ 239Case Study 3: Physiological Levers and Repairs .................................. 241

The Forearm Is Connected to the Elbow Joint.............................. 241Fixing an Intertrochanteric Fracture ............................................. 245Problems ............................................................................................ 247

Notes ........................................................................................................... 248

Contents ix

6 Beam Deflections ............................................................................ 2516.1 Governing Equation ......................................................................... 2516.2 Boundary Conditions .......................................................................2556.3 Solution of Deflection Equation by Integration ............................2566.4 Singularity Functions ...................................................................... 2596.5 Moment Area Method ...................................................................... 2606.6 Beams with Elastic Supports...........................................................2646.7 Strain Energy for Bent Beams ......................................................... 2666.8 Flexibility Revisited and Maxwell-Betti Reciprocal Theorem ... 2696.9 Examples ............................................................................................ 273

Example 6.1 ........................................................................................ 273Solution ................................................................................... 273

Example 6.2 ........................................................................................ 275Solution ................................................................................... 275

Example 6.3 ........................................................................................ 278Solution ................................................................................... 278

Example 6.4 ........................................................................................ 281Solution ................................................................................... 282

6.10 Problems ............................................................................................285Notes ...........................................................................................................288

7 Instability: Column Buckling ...................................................... 2897.1 Euler’s Formula ................................................................................. 2897.2 Effect of Eccentricity ......................................................................... 2947.3 Examples ............................................................................................ 298

Example 7.1 ........................................................................................ 298Solution ................................................................................... 298

Example 7.2 ........................................................................................300Solution ................................................................................... 301

7.4 Problems ............................................................................................303Case Study 4: Hartford Civic Arena ......................................................304Notes ........................................................................................................... 307

8 Connecting Solid and Fluid Mechanics ...................................... 3098.1 Pressure .............................................................................................. 3108.2 Viscosity ............................................................................................. 3118.3 Surface Tension ................................................................................. 3158.4 Governing Laws ................................................................................ 3158.5 Motion and Deformation of Fluids ................................................ 316

8.5.1 Linear Motion and Deformation ......................................... 3168.5.2 Angular Motion and Deformation ..................................... 3178.5.3 Vorticity................................................................................... 3198.5.4 Constitutive Equation (Generalized Hooke’s Law) for Newtonian Fluids ............................................................ 321

8.6 Examples ............................................................................................ 322Example 8.1 ........................................................................................ 322

x Introduction to Engineering Mechanics: A Continuum Approach

Solution ................................................................................... 323Example 8.2 ........................................................................................ 324

Solution ................................................................................... 324Example 8.3 ........................................................................................ 325

Solution ................................................................................... 326Example 8.4 ........................................................................................ 327

Solution ................................................................................... 3278.7 Problems ............................................................................................ 328Case Study 5: Mechanics of Biomaterials ..............................................330

Nonlinearity ...................................................................................... 332Composite Materials ........................................................................333Viscoelasticity....................................................................................336Problems ............................................................................................338

Notes ........................................................................................................... 339

9 Fluid Statics ..................................................................................... 3419.1 Local Pressure ................................................................................... 3419.2 Force Due to Pressure ......................................................................3429.3 Fluids at Rest .....................................................................................3459.4 Forces on Submerged Surfaces .......................................................3489.5 Buoyancy ............................................................................................ 3559.6 Examples ............................................................................................ 357

Example 9.1 ........................................................................................ 357Solution ................................................................................... 357

Example 9.2 ........................................................................................ 358Solution ................................................................................... 359

Example 9.3 ........................................................................................360Solution ................................................................................... 361

Example 9.4 ........................................................................................363Solution ...................................................................................364

Example 9.5 ........................................................................................365Solution ...................................................................................366

9.7 Problems ............................................................................................368Case Study 6: St. Francis Dam ................................................................ 373

Problems ............................................................................................ 375Notes ........................................................................................................... 376

10 Fluid Dynamics: Governing Equations ...................................... 37710.1 Description of Fluid Motion ......................................................... 37710.2 Equations of Fluid Motion ............................................................ 37910.3 Integral Equations of Motion ....................................................... 379

10.3.1 Mass Conservation ...........................................................38010.3.2 F = ma, or Momentum Conservation ............................. 38210.3.3 Reynolds Transport Theorem .........................................385

10.4 Differential Equations of Motion .................................................38610.4.1 Continuity, or Mass Conservation .................................386

Contents xi

10.4.2 F = ma, , or Momentum Conservation .............................38810.5 Bernoulli Equation ........................................................................... 39110.6 Examples ............................................................................................ 392

Example 10.1 ...................................................................................... 392Solution ................................................................................... 393

Example 10.2 ...................................................................................... 394Solution ................................................................................... 395

Example 10.3 ...................................................................................... 396Solution ................................................................................... 397

Example 10.4 ...................................................................................... 398Solution ................................................................................... 399

Example 10.5 ......................................................................................402Solution ...................................................................................402

Example 10.6 ......................................................................................404Solution ...................................................................................405

10.7 Problems ............................................................................................406Notes ...........................................................................................................408

11 Fluid Dynamics: Applications ..................................................... 41111.1 How Do We Classify Fluid Flows? .............................................. 41111.2 What’s Going on Inside Pipes? ..................................................... 41311.3 Why Can an Airplane Fly? ........................................................... 41711.4 Why Does a Curveball Curve? ..................................................... 41911.5 Problems ..........................................................................................423Notes ...........................................................................................................426

12 Solid Dynamics: Governing Equations ...................................... 42712.1 Continuity, or Mass Conservation ...............................................42712.2 F = ma, or Momentum Conservation ..........................................42912.3 Constitutive Laws: Elasticity ........................................................ 431Note.............................................................................................................433

References ................................................................................................ 435

Appendix A: Second Moments of Area ................................................ 439

Appendix B: A Quick Look at the Del Operator ................................. 443Divergence .................................................................................................444

Physical Interpretation of the Divergence .....................................444Example ..............................................................................................445

Curl .............................................................................................................445Physical Interpretation of the Curl .................................................445Examples ............................................................................................446

Example 1 ...............................................................................446Example 2 ...............................................................................446

Laplacian ....................................................................................................447

xii Introduction to Engineering Mechanics: A Continuum Approach

Appendix C: Property Tables ................................................................. 449

Appendix D: All the Equations ............................................................. 455

Index .......................................................................................................... 457

If science teaches us anything, it’s to accept our failures, as well as our successes, with quiet dignity and grace.

Gene Wilder, Young Frankenstein, 1974

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xv

Preface

This book is intended to provide a unified introduction to solid and fluid mechanics and to convey the underlying principles of continuum mechan-ics to undergraduates. We assume that students using this book have taken courses in calculus, physics, and vector analysis. By demonstrating both the connections and the distinctions between solid and fluid mechanics, this book will prepare students for further study in either field or in fields such as bioengineering that blur traditional disciplinary boundaries.

The use of a continuum approach to make connections between solid and fluid mechanics is a perspective typically provided only to advanced under-graduates and graduate students. This book introduces the concepts of stress and strain in the continuum context, showing the relationships between solid and fluid behavior and the mathematics that describe them. It is an introductory textbook in strength of materials and in fluid mechanics and also includes the mathematical connective tissue between these fields. We have decided to begin with the a-ha! of continuum mechanics rather than requiring students to wait for it.

This approach was first developed at Harvey Mudd College (HMC) for a sophomore-level course called “Continuum Mechanics.” The broad, unspe-cialized engineering program at HMC requires that curriculum planners ask themselves, “What specific knowledge is essential for an engineer who may practice, or continue study, in one of a wide variety of fields?” This course was our answer to the question, what engineering mechanics knowledge is essential?

An engineer of any type, we felt, should have an understanding of how materials respond to loading: how solids deform and incur stress; how fluids flow. We conceived of a spectrum of material behavior, with the idealiza-tions of Hookean solids and Newtonian fluids at the extremes. Most mod-ern engineering materials—biological materials, for example—lie between these two extremes, and we believe that students who are aware of the entire spectrum from their first introduction to engineering mechanics will be well prepared to understand this complex middle ground of nonlinearity and viscoelasticity.

Our integrated introduction to the mechanics of solids and fluids has evolved. As initially taught by CLD, the HMC course emphasized the under-lying principles from a mathematical, applied mechanics viewpoint. This focus on the structure of elasticity problems made it difficult for students to relate formulation to applications. In subsequent offerings, JSR chose to embed continuum concepts and mathematics into introductory problems, and to build gradually to the strain and stress tensors. We now establish a “continuum checklist”—compatibility [deformation], constitutive law, and equilibrium—that we return to repeatedly. This checklist provides a frame-work for a wide variety of problems in solid and fluid mechanics.

xvi Introduction to Engineering Mechanics: A Continuum Approach

We make the necessary definitions and present the template for our contin-uum approach in Chapter 1. In Chapter 2, we introduce strain and stress in one dimension, develop a constitutive law, and apply these concepts to the simple case of an axially loaded bar. In Chapter 3, we extend these concepts to higher dimensions, introducing Poisson’s ratio and the strain and stress tensors. In Chapters 4–7 we apply our continuum sense of solid mechanics to problems including torsion, pressure vessels, beams, and columns. In Chapter 8, we make connections between solid and fluid mechanics, introducing properties of fluids and the strain rate tensor. Chapter 9 addresses fluid statics. Applica-tions in fluid mechanics are considered in Chapters 10 and 11. We develop the governing equations in both control volume and differential forms. In Chapter 12, we see that the equations for solid dynamics strongly resemble those we’ve used to study fluid dynamics. Throughout, we emphasize real-world design applications. We maintain a continuum “big picture” approach, tempered with worked examples, problems, and a set of case studies.

The six case studies included in this book illustrate important applica-tions of the concepts. In some cases, students’ developing understanding of solid and fluid mechanics will help them understand “what went wrong” in famous failures; in others, students will see how the textbook theories can be extended and applied in other fields such as bioengineering. The essence of continuum mechanics, the internal response of materials to external loading, is often obscured by the complex mathematics of its formulation. By build-ing gradually from one-dimensional to two- and three-dimensional formu-lations and by including these illustrative real-world case studies, we hope to help students develop physical intuition for solid and fluid behavior.

We’ve written this book for our students, and we hope that reading it is very much like sitting in our classes. We have tried to keep the tone conver-sational and have included many asides that describe the historical context for the ideas we describe and hints at how some concepts may become even more useful later on.

We are grateful to the students who have helped us refine our approach. We are deeply appreciative of our colleague and friend Lori Bassman (HMC)—of her sense of pure joy in structural mechanics and her ability to communi-cate that joy. Lori has been a sounding board, contributor of elegant (and fun) homework problems, and defender of the integrity of “second moment of area” despite the authors’ stubbornly abiding affection for “moment of iner-tia.” We also thank Joseph A. King (HMC), Harry E. Williams (HMC), Josh Smith (Lafayette), James Ferri (Lafayette), Diane Windham Shaw (Lafayette), Brian Storey (Olin), Borjana Mikic (Smith), and Drew Guswa (Smith). We thank Michael Slaughter and Jonathan Plant, our editors at Taylor & Francis/CRC, and their staff.

We want to convey our warmest gratitude to our families. First are Toby, Leda, and Cleo Rossmann. Thanks especially to Toby, for his direct and indi-rect support of this project. And then there’s Joan Dym, Jordana, and Mir-iam, and Matt and Ryan and spouses and partners, and a growing number of grandchildren. We are grateful for their support, love, and patience.

xvii

About the Authors

Jenn Stroud Rossmann is assistant professor of mechanical engineering at Lafayette College. She earned her B.S. and Ph.D. degrees from the University of California, Berkeley. Her current research includes the study of blood flow in vessels affected by atherosclerosis and aneurysms. She has a strong com-mitment to teaching engineering methods and literacy to non-engineers and has developed several courses and workshops for liberal arts majors.

Clive L. Dym is the Fletcher Jones Professor of Engineering Design at Har-vey Mudd College. He earned his B.S. from Cooper Union and his Ph.D. from Stanford University. His primary interests are in engineering design and structural mechanics. He is the author of eleven books and has edited nine others; his two most recent books are Engineering Design: A Project-Based Introduction, 3rd ed. (with Patrick Little, and with Elizabeth J. Orwin and R. Erik Spjut, John Wiley, 2008) and Principles of Mathematical Modeling, 2nd ed. (Academic Press, 2004). Among his awards are the Fred Merryfield Design Award (American Society for Engineering Education [ASEE], 2002) and the Joel and Ruth Spira Outstanding Design Educator Award (American Society of Mechanical Engineers [ASME], 2004). Dr. Dym is a fellow of the ASCE, ASME, and ASEE.